"어떻게?"

I think what one of the most popular phrases of Koreans is "어떻게?" Tell me please how can it translate correctly?

☞ Do you mean "어떻게?" with the question mark? if so, it means
"what do we do?",
"what should I do?", or
"what am I supposed to do?"

Whereas if it is just "어떻게" without the question mark,
it can be translated in both ways which are "HOW" or "WHAT" in English.

How are you doing? = 어떻게 지내?
How do I get there? = 어떻게 거기 갈 수 있어?

What do you think of me? = 절 어떻게 생각하세요?
What are you going to do? = 이제 어떻게 할거야?


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"어떻게?"

I think what one of the most popular phrases of Koreans is "어떻게?" Tell me please how can it translate correctly?
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Yes, 'how' is right.  Normally, it means "How will you do that?" "Which way will you take to do that?"
 Or " Give(tell) me details(explanation) about that"

For example,
A : I have a good idea to serve Korean food to our guests.
B : 어떻게? 
A : We can use catering service. One of my friends told me the foods was nice at reasonable price at her party.


A : Do you know that Sam has a girlfriend.
B : Really? 어떻게?
A : Sam told me he met her at his friend's birthday party. Blah, blah, .....